Trace-Class and Hilbert-Schmidt Operators 2 3. Standard Forms for II1 Factors 5
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L2-Theory of Semiclassical Psdos: Hilbert-Schmidt And
2 LECTURE 13: L -THEORY OF SEMICLASSICAL PSDOS: HILBERT-SCHMIDT AND TRACE CLASS OPERATORS In today's lecture we start with some abstract properties of Hilbert-Schmidt operators and trace class operators. Then we will investigate the following natural t questions: for which class of symbols, the quantized operator Op~(a) is a Hilbert- Schmidt or trace class operator? How to compute its trace? 1. Hilbert-Schmidt and Trace class operators: Abstract theory { Definitions via singular values. As we have mentioned last time, we are aiming at developing the spectral theory for semiclassical pseudodifferential operators. For a 2 S(m), where m is a decaying t 2 n symbol, we showed that Op~(a) is a compact operator acting on L (R ). As a t consequence, either Op~(a) has only finitely many nonzero eigenvalues, or it has infinitely nonzero eigenvalues which can be arranged to a sequence whose norms converges to 0. However, in general the eigenvalues of a compact operator A are non-real. A very simple way to get real eigenvalues is to consider the operator A∗A, which is 1 a compact self-adjoint linear operator acting on L2(Rn). Thus the eigenvalues of A∗A can be list2 in decreasing order as 2 2 2 s1 ≥ s2 ≥ s3 ≥ · · · : The numbers sj (which will always be taken to be the positive one) are called the singular values of A. Definition 1.1. We say a compact operator3 A is a Hilbert-Schmidt operator if !1=2 X 2 (1) kAkHS := sj < +1: j The number kAkHS is called the Hilbert-Schmidt norm (or Schatten 2-norm) of A. -
Math 261Y: Von Neumann Algebras (Lecture 5)
Math 261y: von Neumann Algebras (Lecture 5) September 9, 2011 In this lecture, we will state and prove von Neumann's double commutant theorem. First, let's establish a bit of notation. Let V be a Hilbert space, and let B(V ) denote the Hilbert space of bounded operators on V . We will consider several topologies on B(V ): (1) The norm topology, which has a subbasis (in fact, a basis) of open sets given by fx 2 B(V ): jjx−x0jj < g where x0 2 B(V ) and is a positive real number. (2) The strong topology, which has a subbasis of open sets given by fx 2 B(V ): jjx(v) − x0(v)jj < g, where x0 2 B(V ), v 2 V , and is a positive real number. (3) The weak topology, which has a subbases of open sets given by fx 2 B(V ): j(x(v) − x0(v); w)j < g where x0 2 B(V ), v; w 2 V , and is a positive real number. Each of these topologies is more coarse than the previous. That is, we have norm convergence ) strong convergence ) weak convergence. Warning 1. With respect to the norm topology, B(V ) is a metric space, so its topology is determined by the collection of convergent sequences. However, the strong and weak topologies do not have this property. However, the unit ball of B(V ) is metrizable in either topology if V is a separable Hilbert space. Warning 2. The multiplication map B(V ) × B(V ) ! B(V ) (x; y) 7! xy is not continuous with respect to either the strong or weak topology. -
Preduals for Spaces of Operators Involving Hilbert Spaces and Trace
PREDUALS FOR SPACES OF OPERATORS INVOLVING HILBERT SPACES AND TRACE-CLASS OPERATORS HANNES THIEL Abstract. Continuing the study of preduals of spaces L(H,Y ) of bounded, linear maps, we consider the situation that H is a Hilbert space. We establish a natural correspondence between isometric preduals of L(H,Y ) and isometric preduals of Y . The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements L(H,Y ) in its bidual is automatically a right L(H)-module map. As an application, we show that isometric preduals of L(S1), the algebra of operators on the space of trace-class operators, correspond to isometric preduals of S1 itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of S1 making its multiplication separately weak∗ continuous. 1. Introduction An isometric predual of a Banach space X is a Banach space F together with an isometric isomorphism X =∼ F ∗. Every predual induces a weak∗ topology. Due to the importance of weak∗ topologies, it is interesting to study the existence and uniqueness of preduals; see the survey [God89] and the references therein. Given Banach spaces X and Y , let L(X, Y ) denote the space of operators from X to Y . Every isometric predual of Y induces an isometric predual of L(X, Y ): If Y =∼ F ∗, then L(X, Y ) =∼ (X⊗ˆ F )∗. Problem 1.1. Find conditions on X and Y guaranteeing that every isometric predual of L(X, Y ) is induced from an isometric predual of Y . -
Notes on Von Neumann Algebras
Von Neumann Algebras. Vaughan F.R. Jones 1 October 1, 2009 1Supported in part by NSF Grant DMS93–22675, the Marsden fund UOA520, and the Swiss National Science Foundation. 2 Chapter 1 Introduction. The purpose of these notes is to provide a rapid introduction to von Neumann algebras which gets to the examples and active topics with a minimum of technical baggage. In this sense it is opposite in spirit from the treatises of Dixmier [], Takesaki[], Pedersen[], Kadison-Ringrose[], Stratila-Zsido[]. The philosophy is to lavish attention on a few key results and examples, and we prefer to make simplifying assumptions rather than go for the most general case. Thus we do not hesitate to give several proofs of a single result, or repeat an argument with different hypotheses. The notes are built around semester- long courses given at UC Berkeley though they contain more material than could be taught in a single semester. The notes are informal and the exercises are an integral part of the ex- position. These exercises are vital and mostly intended to be easy. 3 4 Chapter 2 Background and Prerequisites 2.1 Hilbert Space A Hilbert Space is a complex vector space H with inner product h; i : HxH! C which is linear in the first variable, satisfies hξ; ηi = hη; ξi, is positive definite, i.e. hξ; ξi > 0 for ξ 6= 0, and is complete for the norm defined by test jjξjj = phξ; ξi. Exercise 2.1.1. Prove the parallelogram identity : jjξ − ηjj2 + jjξ + ηjj2 = 2(jjξjj2 + jjηjj2) and the Cauchy-Schwartz inequality: jhξ; ηij ≤ jjξjj jjηjj: Theorem 2.1.2. -
Algebras Have Amenable Unitary Groups
PROCEEDINGSOF THE AMERICANMATHEMATICAL SOCIETY Volume 114, Number 3, MARCH 1992 NUCLEAR C*-ALGEBRASHAVE AMENABLE UNITARY GROUPS ALAN L. T. PATERSON (Communicated by Paul S. Muhly) Abstract. Let A be a unital C*-algebra with unitary group G. Give G the relative (Banach space) weak topology. Then G is a topological group, and we show that A is nuclear if and only if there exists a left invariant mean on the space of right uniformly continuous, bounded, complex-valued functions on G. Nuclear C*-algebras and injective von Neumann algebras are of fundamental importance in operator algebra theory. The two classes of algebras are related through the remarkable result (of Choi-Effros and Connes): a C*-algebra A is nuclear if and only if A** is an injective von Neumann algebra. From the work of Haagerup [3], nuclearity is the same as amenability (in the Banach algebra sense) for C*-algebras. Further, from the deep work of Connes and others, it is known that injectivity, Property P, hyperfiniteness and amenability are all equivalent for a von Neumann algebra M. The relationship between injectivity and classical amenability for topological groups is established in a result of de la Harpe [4] discussed below. We note here that Haagerup in [3] proves that the injectivity of M is equivalent to the existence of a left invariant mean on a certain space of functions on the isometry semigroup of M. The author plans to discuss the relationship between the invariant mean results of Haagerup and de la Harpe in a future paper. We recall some notions from topological group theory. -
S0002-9947-1980-0570783-0.Pdf
transactions of the american mathematical society Volume 260, Number 1, July 1980 NONSTANDARD EXTENSIONS OF TRANSFORMATIONS BETWEEN BANACH SPACES BY D. G. TACON Abstract. Let X and Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an Nj -saturated enlargement by X and Y respectively. If 9> (X, Y) denotes the space of bounded linear transformations then a subset S of elements of ® (X, Y) extends naturally to a subset S of *$>(X, Y). This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that (® (X, Y))~ c 'S (X, Y) provided X and Y are infinite dimensional and contrasts this with the inclusion %(H) c (%(H))~ where %(H) denotes the space of compact operators on a Hubert space. The nonstandard hull of a uniform space was introduced by Luxemburg [14] and the construction, in the case of normed spaces, has since been investigated in a number of papers (see, for example [4], [6], [7], [8], [9], [10], [15]). Our intention here is to examine the behaviour of single bounded linear transformations and classes of bounded linear transformations under the nonstandard hull extension. We recount the principal ideas. Consider a normed space (X, || • ||) embedded in a set theoretical structure 911 and consider an N, -sa tura ted enlargement *91t of 9H. An element p G *X is said to be finite if *||p|| is a finite element of *R, the set of such elements being denoted by fin(*Ar). -
Note on Operator Algebras
Note on Operator Algebras Takahiro Sagawa Department of Physics, The University of Tokyo 15 December 2010 Contents 1 General Topology 2 2 Hilbert Spaces and Operator Algebras 5 2.1 Hilbert Space . 5 2.2 Bounded Operators . 6 2.3 Trace Class Operators . 8 2.4 von Neumann Algebras . 10 2.5 Maps on von Neumann Algebras . 12 3 Abstract Operator Algebras 13 3.1 C∗-Algebras . 13 3.2 W ∗-algebras . 14 1 Chapter 1 General Topology Topology is an abstract structure that can be built on the set theory. We start with introducing the topological structure by open stets, which is the most standard way. A topological space is a set Ω together with O, a collection of subsets of Ω, satisfying the following properties: ∙ 휙 2 O and Ω 2 O. ∙ If O1 2 O and O2 2 O, then O1 \ O2 2 O. ∙ If O훼 2 O (훼 2 I) for arbitrary set of suffixes, then [훼2I O훼 2 O. An element of O is called an open set. In general, a set may have several topologies. If two topologies satisfy O1 ⊂ O2, then O1 is called weaker than O2, or smaller than O2. Topological structure can be generated by a subset of open spaces. Let B be a collection of subsets of a set Ω. The weakest topology O such that B ⊂ O is called generated by B. We note that such O does not always exist for an arbitrary B. Figure 1.1: An open set and a compact set. We review some important concepts in topological spaces: 2 ∙ If O is an open set, then Ω n O is called a closed set. -
On the Characterization of Trace Class Representations and Schwartz Operators
On the characterization of trace class representations and Schwartz operators Gerrit van Dijk, ∗ Karl-Hermann Neeb, y Hadi Salmasian,z Christoph Zellner x January 3, 2016 Abstract In this note we collect several characterizations of unitary representa- tions (π; H) of a finite dimensional Lie group G which are trace class, i.e., for each compactly supported smooth function f on G, the operator π(f) is trace class. In particular we derive the new result that, for some m 2 N, m all operators π(f), f 2 Cc (G), are trace class. As a consequence the cor- responding distribution character θπ is of finite order. We further show π is trace class if and only if every operator A, which is smoothing in the sense that AH ⊆ H1, is trace class and that this in turn is equivalent to the Fr´echet space H1 being nuclear, which in turn is equivalent to the re- alizability of the Gaussian measure of H on the space H−∞ of distribution vectors. Finally we show that, even for infinite dimensional Fr´echet{Lie groups, A and A∗ are smoothing if and only if A is a Schwartz operator, i.e., all products of A with operators from the derived representation are bounded. Introduction Let (π; H) be a (strongly continuous) unitary representation of the (possibly infinite dimensional) Lie group G (with an exponential function). Let H1 be its subspace of smooth vectors. On this space we obtain by d dπ(x)v = π(exp tx)v dt t=0 the derived representation of g which we extend naturally to a representation of the enveloping algebra U(g), also denoted dπ. -
Trace Class Conditions for Functions of Schrödinger Operators
Commun. Math. Phys. 335, 477–496 (2015) Communications in Digital Object Identifier (DOI) 10.1007/s00220-014-2205-8 Mathematical Physics Trace Class Conditions for Functions of Schrödinger Operators Rupert L. Frank1, Alexander Pushnitski2 1 Mathematics 253-37, Caltech, Pasadena, CA 91125, USA. E-mail: [email protected] 2 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK. E-mail: [email protected] Received: 4 February 2014 / Accepted: 26 May 2014 Published online: 6 November 2014 – © The Author(s) 2014 Abstract: We consider the difference f (−+V )− f (−) of functions of Schrödinger operators in L2(Rd ) and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions f and in V belonging only to some L p space. This is motivated by applications in mathematical physics related to Lieb–Thirring inequalities. We show that in the particular case of Schrödinger operators the well-known sufficient conditions on f , based on a general operator theoretic result due to V. Peller, can be considerably relaxed. We prove similar theorems for f (− + ) − (−) − d (− α )| V f dα f + V α=0. Our key idea is the use of the limiting absorption principle. 1. Introduction and Main Results 1.1. Setting of the problem. In this paper we consider functions f (H) and f (H0) of the perturbed and unperturbed Schrödinger operators 2 d H =− + V, H0 =− in L (R ) (1.1) and we investigate which assumptions on the real-valued potential V and on the function f guarantee the property that f (H) − f (H0) ∈ S1 (1.2) or d f (H) − f (H ) − f ((1 − α)H + αH)|α= ∈ S , (1.3) 0 dα 0 0 1 Copyright© 2014 by the authors. -
Proved That Every Real Separable Banach Space Contains a Separable Hilbert Space As a Dense Embedding, and This Space Is the Support of a Gaussian Measure
. SOME BANACH SPACES ARE ALMOST HILBERT TEPPER L. GILL1∗ AND MARZETT GOLDEN1 Abstract. The purpose of this note is to show that, if B is a uniformly convex Banach, then the dual space B0 has a \ Hilbert space representation" (defined in the paper), that makes B much closer to a Hilbert space then previously suspected. As an application, we prove that, if B also has a Schauder basis (S- basis), then for each A 2 C[B] (the closed and densely defined linear operators), there exists a closed densely defined linear operator A∗ 2 C[B] that has all the expected properties of an adjoint. Thus for example, the bounded linear operators, L[B], is a ∗algebra. This result allows us to give a natural definition to the Schatten class of operators on a uniformly convex Banach space with a S-basis. In particular, every theorem that is true for the Schatten class on a Hilbert space, is also true on such a space. The main tool we use is a special version of a result due to Kuelbs [K], which shows that every uniformly convex Banach space with a S-basis can be densely and continuously embedded into a Hilbert space which is unique up to a change of basis. 1. Introduction In 1965, Gross [G] proved that every real separable Banach space contains a separable Hilbert space as a dense embedding, and this space is the support of a Gaussian measure. This was a generalization of Wiener's theory, that was based on the use of the (densely embedded Hilbert) Sobolev space H1[0; 1] ⊂ C[0; 1]. -
KERNELS of TRACE CLASS OPERATORS CHRIS BRISLAWN (Communicated by John B
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 104, Number 4, December 1988 KERNELS OF TRACE CLASS OPERATORS CHRIS BRISLAWN (Communicated by John B. Conway) ABSTRACT. Let X C Rn and let K be a trace class operator on L2(X) with corresponding kernel K(x,y) € L2(X x X). An integral formula for tr K, proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of K into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be sufficient for positive operators. The proofs make use of the boundedness of the Hardy- Littlewood maximal function on L2(R"). 1. Introduction. Given a kernel function, K(x,y), what are necessary and sufficient function-theoretic conditions on K(x, y) so that the corresponding integral operator on L2(X) is traceable, and how does one compute tr A in terms of the given kernel? These questions arise quite naturally in, for example, the theory of unitary group representations. For the case when X C Rn, the main result of this paper, Theorem 3.1, answers the second question completely by proving a generalization of the formula trK= K(x,x)dx that holds for arbitrary trace class kernels, and we show how this result can be related to the factorization of a trace class operator into a product of Hilbert- Schmidt operators. Theorem 3.1 also provides a partial answer to the first question by proving that a Hilbert-Schmidt operator, K, is traceable only if / MK(x, x) dx < oo, where MK is the Hardy-Littlewood maximal function of the kernel K(x,y). -
Arxiv:1609.01093V2 [Math.OA] 1 Oct 2016 That Tv Ccp)Mp.Tecase the Maps
FINITE DIMENSIONAL APPROXIMATION PROPERTIES OF C∗-MODULES MASSOUD AMINI Abstract. We study nuclearity and exactness for module maps on C∗-algebras which are C∗-module over another C∗-algebra with compatible actions and study finite dimensional approximation properties of such C∗-modules. We prove module versions of the results of Kirchberg and Choi-Effros. As a con- crete example we extend the finite dimensional approximation properties of reduced C∗-algebras and von Neumann algebras on discrete groups to these operator algebras on inverse semigroups with the module structure coming from the action of the C∗-algebras on the subsemigroup of idempotents. 1. introduction Finite dimensional approximation properties of C∗-algebras is core subject in modern theory of operator algebras [4]. These include important notions such as nuclearity, exactness and weak expectation property (WEP). The results in this direction are obtained based on the classical extension and dilation results due to Arveson, Wittstock and Stinespring. Some of these results are also valid for C∗- module maps [20]. It is desirable then to consider the finite dimensional approxi- mation properties of C∗-modules. The motivation is two fold: A finite dimensional approximation scheme for C∗-morphisms is as follows: θ / A ● ;/ B ●● ✇✇; ●●ϕn ψn ✇✇ ●● ✇✇ ●# ✇✇ M kn(C) ∗ where A and B are C -algebras and ϕn and ψn are contractive completely pos- itive (c.c.p.) maps. The case A = B and θ = idA is of special interest. There are situations that such an approximate decomposition is needed through arXiv:1609.01093v2 [math.OA] 1 Oct 2016 M ∗ kn (N) for a C -algebra or von Neumann algebra N.